Question: The sum of the lengths of the twelve edges of a rectangular box is $140$, and the distance from one corner of the box to the farthest corner is $21$. What is the total surface area of the box?
Solution: Let $a$, $b$, and $c$ be the dimensions of the box. It is given that \[140=4a+4b+4c{\qquad \rm and
\qquad}21=\sqrt{a^2+b^2+c^2}\] hence \[35=a+b+c{\qquad (1)\qquad
\rm and\qquad}441=a^2+b^2+c^2{\qquad (2)}.\]

Square both sides of $(1)$ and combine with $(2)$ to obtain \begin{align*}
1225 & = (a+b+c)^2 \\
&= a^2+b^2+c^2+2ab+2bc+2ca \\
&= 441+2ab+2bc+2ca.
\end{align*}

Thus the surface area of the box is \[ 2ab+2bc+2ca=1225-441=\boxed{784}.\]